2021/01/03 by Ivan Chajda, Chajda, Ivan, Helmut Länger +1
Mathematics · #03G10 #03G25 #03G47 #06D30 #FOS: Mathematics #Rings and Algebras (math.RA) #math.RA #msc:03G10 #msc:03G25 #msc:03G47 #msc:06D30
paper · pdf · doi:10.48550/arxiv.2101.00677
arxiv created 2021/01/03 · arxiv updated 2021/01/05
Given an integral commutative residuated lattice L=(L,\vee,\wedge), its full twist-product (L2,\sqcup,\sqcap) can be endowed with two binary operations \odot and ⇒ introduced formerly by M. Busaniche and R. Cignoli as well as by C. Tsinakis and A. M. Wille such that it becomes a commutative residuated lattice. For every a in L we define a certain subset Pa(L) of L2. We characterize when Pa(L) is a sublattice of the full twist-product (L2,\sqcup,\sqcap). In this case Pa(L) together with some natural antitone involution ' becomes a pseudo-Kleene lattice. If L is distributive then (Pa(L),\sqcup,\sqcap,') becomes a Kleene lattice. We present sufficient conditions for Pa(L) being a subalgebra of (L2,\sqcup,\sqcap,\odot,⇒) and thus for \odot and ⇒ being a pair of adjoint operations on Pa(L). Finally, we introduce another pair \odot and ⇒ of adjoint operations on the full twist-product of a bounded commutative residuated lattice such that the resulting algebra is a bounded commutative residuated lattice satisfying the double negation law and we investigate when Pa(L) is closed under these new operations \odot and ⇒.